The Alhambra is far and away Granada's most famous landmark. Built in the 14th century and restored to its present glory in the 19th century, this Moorish fortress is known as one of the true architectural masterpieces left behind by Muslim rule in Spain. The sprawling castle consists of a vast number of buildings and is worth a day's exploration of its pools, gardens, ramparts, and red clay towers. Highlights include the Charles V Palace with its unique circular patio, the Comares Palace (which was once the official residence of the king), and the oldest structure in the complex, the 9th century fortress known as the Alcazaba. Also, if there are enough hours in your day, be sure to visit the adjacent Generalife, a sprawling, 700 year old complex that was once the home of the Nasrid Emirs.
Ticket prices are mighty steep (49 euro/about 70 US dollars), but they can get away with it because the Alhambra is by far the biggest reason why travelers come to Granada. Pay the money and cherish the memories. Also be sure to purchase your tickets ahead of time to avoid waiting to buy them; the weekends can get incredibly busy in peak season.